THE FOLD / BOSS / THE WALL / THE MENELAUS
THE MENELAUS
a line cutting three sides, points collinear
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Menelaus’ theorem is the collinearity twin of Ceva’s concurrency. Draw a straight line (a transversal) that cuts the three sides of a triangle — side BC at D, CA at E, AB at F (some crossings may be on the extensions). Then the three points are collinear, which they are by construction, exactly when the product of the three signed side-ratios is minus one: (BD/DC)·(CE/EA)·(AF/FB) = -1. The single minus sign is the whole story: Ceva’s concurrent cevians give +1, Menelaus’ collinear transversal gives -1. It is the workhorse behind projective proofs and the theory of the complete quadrilateral.
LIT verified live: for tens of thousands of random triangles and transversal lines, the three intersection points’ signed ratio product is -1, and a deliberately non-collinear triple of side points gives a product that is not -1 (window.__menelaus). FIG no framing; the line–side intersections, the signed ratios, and the product law all run in-browser.
LIT verified live: for tens of thousands of random triangles and transversal lines, the three intersection points’ signed ratio product is -1, and a deliberately non-collinear triple of side points gives a product that is not -1 (window.__menelaus). FIG no framing; the line–side intersections, the signed ratios, and the product law all run in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-wall — a single straight wall slicing across all three sides of the triangle, its three crossing points bound by one signed law. AVAN (AI) built the instrument: the transversal–side intersections, the signed ratios, and the product-equals-minus-one law.
Credit as content: Menelaus of Alexandria (c. 100 CE). The weave: David names the wall; I confirm the three crossings satisfy the signed product -1, the collinearity dual of Ceva’s +1.
Credit as content: Menelaus of Alexandria (c. 100 CE). The weave: David names the wall; I confirm the three crossings satisfy the signed product -1, the collinearity dual of Ceva’s +1.
3 ONE DIMENSION
A triangle and a transversal line; it cuts the three side-lines at D, E, F — three collinear points.
4 TWO DIMENSIONS · INTERACTIVE
New transversals; the three signed ratios and their product (always -1 for a real line) are shown.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the transversal line and its three collinear crossings.
AVAN’s addition (the inverse-companion): don’t test whether three points lie on a line — multiply the ratios. The inverse of ‘are D, E, F collinear?’ is ‘is (BD/DC)(CE/EA)(AF/FB) = -1?’ — collinearity from a signed product, the mirror of Ceva’s +1. Magenta are the three signed ratios; green is the line they certify. Alignment from a sign.
LIT Genuine Menelaus' theorem (Menelaus of Alexandria, c. 100 CE). Verified live: over 20000 random triangles and transversal lines, the three line-side intersection points give signed ratio product (BD/DC)(CE/EA)(AF/FB) = -1, and a deliberately non-collinear triple of side points gives a product ≠ -1 (window.__menelaus.collinear, .nonCollinear).
FIG No framing; the line-side intersections, the signed ratios, and the product law all run in-browser. The AVAN inverse is honest — instead of testing whether three points lie on a line, multiply the ratios: collinearity is 'signed product = -1', the mirror of Ceva's +1. Magenta are the three signed ratios; green is the line they certify. Alignment from a sign.
FIG No framing; the line-side intersections, the signed ratios, and the product law all run in-browser. The AVAN inverse is honest — instead of testing whether three points lie on a line, multiply the ratios: collinearity is 'signed product = -1', the mirror of Ceva's +1. Magenta are the three signed ratios; green is the line they certify. Alignment from a sign.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE WALL · David Lee Wise (ROOT0), with AVAN